You yourself proved P2 true — fishfry
But you just proved P2 yourself! You agreed that under the hypothesis of being able to recite a number at successively halved intervals of time, there is no number that is the first to not be recited. — fishfry
I have agreed repeatedly that we can't "count all the natural numbers backwards" since an infinite sequence has no last element. — fishfry
a. I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum
b. I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum
Here is our premise:
P1. In both (a) and (b) there is a bijection between the series of time intervals and the series of natural numbers and the sum of the series of time intervals is 60.
However, the second supertask is metaphysically impossible. It cannot start because there is no largest natural number to start with. Therefore, P1 being true does not entail that the second supertask is metaphysically possible.
Therefore, P1 being true does not entail that the first supertask is metaphysically possible.
Given P2, what is the first natural number not recited? I seem to remember having asked you this several times already. — fishfry
People disagree with the premise because we are not confident we can use such intuitions when the — unintuitive — concept of infinity is involved. — Lionino
The fallacy in his reasoning is that it does not acknowledge that for all tn >= t1/2 the lamp is on iff the lamp was off and I pressed the button to turn it on and the lamp is off iff the lamp was on and I pressed the button to turn it off.
True. And that implies time is discrete how? — Lionino
I'm not trying to find a solution, just to understand what's going on. Not so much why it's wrong, but why anyone would think it was right. Where does the illusion come from? — Ludwig V
It isn't a physically possible task. — noAxioms
Though I don't quite see how your B2 follows from your B1. — Ludwig V
You mean that we don't know the state of X at the last step before t(1), even though X must have been in one state or the other? — Ludwig V
This puzzles me. Is this t(1) the same t as the t(1) in C3? It can't be. There must be a typo there somewhere. — Ludwig V
One question, then - The state of X at any t(n), depends on its predecessor state at t(n-1), doesn't it? Isn't that a definition? Why is it inapplicable to t(1)? — Ludwig V
I have wondered whether one could replace the Thompson lamp with a question, such as whether the final number was odd or even. That would work if you start with an odd divisor and don't express everything in decimals. Perhaps it would work for all examples if you ask whether the number of steps taken is odd or even when the minute is up. I think. — Ludwig V
The lack of a first step does not prevent the beginning of the task — noAxioms
I described exactly how to do that — noAxioms
Ok. Just talking about standard mathematical sequences. It's a common misunderstanding in this thread. The sequence 1/2, 3/4, 7/8, ... has a limit, namely 1, but no last element.
The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of Michael's example of counting backwards. — fishfry
Given your reluctance to clarify the definition of the verb 'to start', I cannot respond appropriately to this statement. I gave a pair of options, or you can supply your own, so long as it isn't open to equivocation. — noAxioms
I've repeatedly challenged you to name the first number not verbalized when we count forward 1, 2, 3, ... at successively halved intervals of time. — fishfry
You (as well as Meta above) seem to insist on an additional premise of the necessity of a bound to something explicitly defined to be unbounded. — noAxioms
Good! Then it's logically possible for it to. An infinite number of things can complete without blowing up logic. — fdrake
How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down. — noAxioms
So when I hear Michael talking about the impossibility of a geometric series "completing" (so to speak) due to being unable to recite the terms in finite time... — fdrake
Repeating yourself three times, while ignoring my responses, does not help further the conversation. — Bob Ross
This is false; and does not follow from the former claim you made. — Bob Ross
Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility! — fdrake
Which would be odd, seeing as such an object has a model in set theory — fdrake
I think you're missing Bob Ross's point.
A belief that "aliens exist" is not the same as a belief about the proposition "I believe that aliens exist" — ChrisH
Ok. I asked for a reference. Now I have no idea what I'm supposed to conclude from this. — fishfry
Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? It's not something that we can just assume unless proven otherwise. Even Benacerraf in his criticism of Thomson accepted this. — Michael
Feel free to give a reference, else I can't respond. — fishfry
I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his.
you are thinking that "aliens exist" is true or false relative to a belief — Bob Ross
Yes that is a proposition, and whether or not it is true or false is independent of any belief about it — Bob Ross
Might show it's logically possible tho. — fdrake
By providing a standard mathematical object which is infinite, has no final element, tends to an end state, and has an infinite number of occurrences ("steps"), but occurs in finite time. — fdrake
A clock ticks 1 time per second.
You start with a cake.
Every second the clock ticks, cut the cake in half.
Make the clock variable, it ticks n times a second.
The limit clock as n tends to infinity applies an infinity of divisions to the cake in 1 second. There is no final operation.
There's nothing logically inconsistent in this, it's just not "physical". — fdrake
I don't see that. At best he showed that one example is undefined. To prove something impossible it must be shown that there is not a single valid one. To prove them physically possible, one must show only a single case (the proverbial black swan). Nobody has done either of those (not even Zeno), so we are allowed our opinions. — noAxioms
A. At t0 the lamp is off, at t1/2 I press the button, at t3/4 I press the button, at t7/8 I press the button, and so on ad infinitum
Compare with:
B. At t0 the lamp is off, at t1/2 I press the button
The status of the lamp at t1 must be a logical consequence of the status of the lamp at t0 and the button-pressing procedure that occurs between t0 and t1 because nothing else controls the behaviour of the lamp.
If no consistent conclusion can be deduced about the lamp at t1 then there’s something wrong with your button-pressing procedure.
So the fact that the status of the lamp at t1 is "undefined" given A is the very proof that the supertask described in A is impossible. — Michael
and all you have done is taking a claim that I am obviously going to deny — Bob Ross